Topics in String Theory | Lecture 3
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Part Number | 3 | |
| Number of Parts | 9 | |
| Author | ||
| License | CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/15114 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
1
2
3
4
5
6
7
8
9
00:00
Voltaic pileMitsubishi A6M ZeroVideoRadiationMassHeatGas turbineStellar atmosphereMeasurementHot workingSolar energyLine-of-sight propagationTauNetztransformatorAngle of attackCosmic distance ladderAC power plugs and socketsYearUniverseBook designForgingFlatcarPlanck unitsTemperaturePhotonDirect currentLuminosityLocherEffects unitField-effect transistorMagnetspuleCapital shipKickstandTypesettingAtmospheric pressureWavelengthContinuous trackThermalGasPaperRoots-type superchargerStarEnergiesparmodusAngeregter ZustandQuantum fluctuationAmmeterCartridge (firearms)SizingRegentropfenRulerLambda baryonShip breakingDrehmassePressureFuelLastBlackHauptsatz der Thermodynamik 2RocketElektronenkonfigurationAutumnFlugbahnKilogramLightGround stationJet (brand)Control panel (engineering)BackpackPower (physics)Water vaporHose couplingCogenerationSpare partAccelerationBlack holeDensityTrajectoryBird vocalizationNatürliche RadioaktivitätVeränderlicher SternSunlightSpecific weightSpeed of lightSkyViseRoll formingAlcohol proofA Large Ion Collider ExperimentOrder and disorder (physics)Differential (mechanical device)Cylinder headMinuteNewton's laws of motionEngine displacementGlobal warmingDesertionMicrowaveVolumetric flow rateForceNear field communicationFoot (unit)AM-Herculis-SternKosmischer StaubHot isostatic pressingPhotonicsClassical mechanicsWärmekapazitätCosmic microwave background radiationThermodynamic equilibriumNoise figureStandard cellMechanicGlitchMountainRemotely operated underwater vehicleParticle physicsEraserFrictionTiefdruckgebietSpantFACTS (newspaper)Crystal structureEnergy levelAerodynamicsTool bitNegationMicroscopeStellaratorFinger protocolBasis (linear algebra)Reference workPerturbation theoryFundamental frequencyShort circuitAnalog signalAtomAtmosphärische TurbulenzSingle (music)ThermometerYachtHourCableKopfstützeGlasflussShip classNightGunDigging stickPlant (control theory)Violet (color)HitzeschildHall effectElektronenbeugungWeightBottleSchwache LokalisationTurningWatercraft rowingCombined cycleHalo (optical phenomenon)Summer (George Winston album)BauxitbergbauKey (engineering)GentlemanPhotographyWhiteCargo shipPipingBrickyardMagnetic coreMagic (cryptography)Penumbra: OvertureShipwreckPelzwareRadioactive decayCoatingGirl (band)CoalPrinter (publishing)GameScrewdriverBuoy tenderMultiplizitätCardinal directionRolling (metalworking)Cell (biology)Surround soundStock (firearms)EisengießereiStormFermi, EnricoRailroad carFlavour (particle physics)AbsorbanceFlip (form)StationeryCardboard (paper product)WednesdayMarch (territory)Filter (optics)Photonic crystalColorfulnessSunriseFocus (optics)Ballpoint penProzessleittechnikMorales Giles MariscalVolatility (chemistry)Vehicle armourSource (album)TowingWind farmAccess networkPlane (tool)ToolElectronic mediaPickup truckQuarkEnergy storageSoundCar seatFood storageBombFunkgerätSeries and parallel circuitsNyquist stability criterionAbsorption (electromagnetic radiation)Group delay and phase delayPagerCluster (physics)WeekBand gapMonthScale (map)Cooling towerRopeMatClockCocktail party effectHochfrequenzübertragungMolding (process)DecemberGlättung <Elektrotechnik>WatchAtmosphere of EarthSensorWireAstronomisches FensterLeadBulletin boardFiling (metalworking)Mint-made errorsBook coverKette <Zugmittel>Satellite
Transcript: English(auto-generated)
00:05
Stanford University. I think I mentioned last time that the horizon of a black hole is not a place where anything very dangerous is
00:21
happening, a place where anything very traumatic would happen to somebody who fell in. In fact, if you take a big black hole, now how big? Let's say really big, really big black hole. What would you find if you fell through the horizon of the
00:42
black hole? The answer is you would find nothing there. It would look exactly like flat space or almost exactly like flat space if the horizon was big enough. And it would just look like empty space. Well, that's a little odd because the mathematics that we
01:00
wrote down looks like it does have something striking going on at the horizon. And tonight I want to resolve that. There's a little bit of mathematics. Mathematics is about at the edge of what I do in this class. It's not terribly hard, but we're going to follow through
01:23
and unravel what happens very, what the horizon of the black hole is really like. What's going on there? All right. So let's write down the metric of the black hole.
01:46
That's the beginning always for the starting point for black holes is to write down a metric. I usually write D tau squared where tau stands for proper time. I'm going to use the other convention.
02:00
The other convention is just to change the sign of the whole thing and call it proper distance. I'll explain in a moment. Proper distance squared is just the negative of the proper time between two points. I'm just doing that partly for variety, partly because it
02:23
will keep me sane because I know all of these formulas in this form. Remember that what we wrote for the tau squared was just something involving D t squared, right? D t squared. In flat space, in fact, it was just D t squared minus D
02:40
x squared plus D y squared plus D z squared. That was D tau squared. Notice that it's positive if D t is bigger than D tau. So for a longer trajectory, the time difference is bigger than the space difference. For example, if the object was standing perfectly still in
03:02
space, then D x would be absolutely zero. We've defined the proper time, D tau squared, so that it would be positive. So that the square of it would be positive. But supposing we were studying a space-like distance, the distance between two points at a fixed time.
03:21
Then, with the original definition, this would be negative. And when we took the square root of it, we would say that the distance between these two points is imaginary. Well, we're going to get stuck one way or another. If we define the invariant distance so that the square of
03:43
it is positive for time-like things, then it's going to be negative for space-like things and vice versa. So what people do is to use two different conventions. Just whenever you don't like your convention, you change the sign. Well, now's the time that I don't like my convention. So I'm going to change the sign and define D x squared.
04:03
What does S stand for? Spatial distance, rather than time distance. But it's just the negative of the proper time, or the square of the proper time. It's not a big deal. But let's write down the metric of a Schwarzschild black
04:22
hole in this notation. It's 1 minus 2 m g over R. Sorry, what am I doing here? The S squared has a minus and then a plus. We change the sign from the proper time.
04:42
That's the spatial interval, the opposite. A minus here, dt squared. And then, you remember what goes here? 1 over 1 minus 2 m g over R, dr squared.
05:03
And then, another piece here, which I'm going to ignore. This had to do with the angles on the sky. It was R squared times something I think I called the omega squared. But let's ignore it for now. Let's not worry about angular distance on the sky. Let's just worry about time and radial distance. All right, so plus a bit more here.
05:25
When you're far from the black hole, that means when R is large, large compare, what's 2 m g called? It's the Schwarzschild radius. It's the Schwarzschild radius of the black hole. 2 m g, if I were to put in speeds of light, it would
05:41
be 2 m g over c squared. So right now, c is equal to 1. 2 m g is the Schwarzschild radius. And if you're far away from the black hole, R being much bigger than the Schwarzschild radius, then this is negligible. This is negligible. And it's just dt squared minus dt squared plus dr
06:04
squared, exactly as it would be in flat space. Plus a bit more. All right, as you get close to the black hole, you come to the point where R is approximately 2 m g. In other words, a distance more or less the same as the
06:20
Schwarzschild radius. And then something rather dramatic happens to these coefficients. This one becomes zero. And this one, in the denominator, it gets to be zero, but a zero in the denominator means something's infinite. Sure, it sounds like something unusual is happening.
06:40
All right, let me show you exactly what's happening there. And it's a great big nothing. Nothing is happening there. It's an artifact of the choice of coordinates. I'm going to show you some examples. And then we'll work out in detail what's happening at the horizon of the black hole.
07:04
Over here, I want to discuss coordinates. And for the moment now, I'm only talking about ordinary Cartesian coordinates on the plane, on the blackboard. Let's call them Y and X.
07:20
Later on, the vertical axis will become the time axis. But for now, the vertical axis is just the Y axis. All right, now what's the metric of the XY plane? That's pretty easy. That's just the S squared is equal to the X squared plus the Y squared.
07:40
And nothing takes place especially interesting at the origin or any place else. Now, let's go to polar coordinates. Polar coordinates means we characterize a point by a distance which I won't call R because I don't want to confuse it with this R over here.
08:01
It's a different thing. Let's call it rho, the Greek letter rho, which is sort of like R. And what's the other variable? Theta, angle. Okay, first of all, what is X in terms of rho and theta?
08:27
We'll just drop a perpendicular. X is equal to rho times cosine of theta, right? That's the definition of cosine. And Y is rho sine theta.
08:43
Everybody agree? Okay, good. Now, what about the metric in terms of rho and theta? Supposing we start at this point over here and we make a little excursion over to here, a little differential displacement. And that little differential displacement corresponds to
09:01
changing R a certain amount. Let's take the red line here is dr. That's dr. And the other red line here, how long is that? This distance here is just dr.
09:21
But what about the distance perpendicular to it? Is it d theta? Rho d theta. Rho d theta. The further away you are, the bigger a given d theta corresponds to. So that's rho d theta. So now, can you write down what the metric is?
09:42
Sure you can. ds squared is equal to dr squared. d rho squared. Sorry, this is d rho. d rho, not the R. We save R for over there.
10:01
d rho, d rho squared plus rho squared d theta squared. That's the metric in polar coordinates. Notice it has a funny property. The funny property, nothing gets infinite in this metric. But notice it's a funny property that the coefficient of
10:23
the d theta becomes zero at rho equals zero. That means moving in theta, you don't get anywhere. You don't move, even if you move in theta. Well, why not? Well, because you're just not moving very much when you move around theta at the origin.
10:42
Here's a situation. Also, this becomes zero. And it looks like when t changes, nothing is happening if this is zero. We'll see as we go along that they're very, very similar kinds of things. Okay, everybody understand that?
11:01
That's just good old polar coordinates. But now, let's go to Minkowski space, the space time of special relativity. Instead of using y for the vertical axis, we will now use, let's call it capital T. I'm not going to call it small t.
11:22
I'm going to call it capital T. And the horizontal axis, I will call x again. All right, so t is replacing y now. And what's the metric now? The metric now is ds squared is equal to minus dt squared
11:43
plus dx squared. This is just two dimensional now. We don't need the other dimensions for this argument to explain what's happening here. Just minus dt squared plus dx squared. The minus is there for special relativity. So, that's our metric.
12:01
It looks sort of like this, but only a sign change. All right, I'm going to now introduce polar coordinates, but they're called hyperbolic polar coordinates. Let's just draw in here the light cones. These are the light cones that light rays move along
12:21
that go through the origin. We're going to introduce, again, a radial variable or a rho, which measures distance from the origin, but not ordinary distance. Proper distance from the origin. And a time variable that I'm going to call omega.
12:43
Here's the relationship. Here's the definitions. X, I go up to the top there. X equals rho cosine theta. We're going to replace that by x equals rho hyperbolic cosine theta, but not theta.
13:01
We'll call it omega. It's a standard terminology for a hyperbolic angle. I'm going to tell you more about hyperbolic angles as we go along, if you've forgotten. But let's just write down the equations on the blackboard. But this isn't right. I left out something. What did I leave out? Rho.
13:21
Completely parallel to what I've written up there. And t is equal to rho times the hyperbolic sine of omega. All right. Now what are hyperbolic sine and cosine? Well, whatever they are, let's come back to cosine and sine. Cosine squared plus sine squared equals 1.
13:45
That's a statement of basically Pythagorean theorem, that this side squared plus that side squared is that side squared is the hypotenuse squared for a unit triangle. How about for these hyperbolic guys? What's the equation for them?
14:04
Cos squared omega minus sine squared omega equals 1. All right, that's one thing. And let me tell you what cos and sine mean. From the diagram up above, we can think of this by first
14:24
drawing a circle. By first drawing a unit circle, draw a unit circle, draw the angle theta, and then x is given in terms of cosine. For a unit circle, it's just x is equal to cosine,
14:42
y is equal to sine. We do the same thing, except not with a circle, but with a hyperbola. We take the hyperbola, let's raise this up. Let's take the hyperbola, not x squared plus y squared equals 1, but x squared minus t squared equals 1.
15:03
So take the hyperbola, x squared minus t squared equals 1. That looks like this. That's a hyperbola that looks like this. And now take a point on that unit hyperbola, any point,
15:22
and draw the same kind of right triangle that we drew for the circle. Take a point, drop the perpendicular, drop the perpendicular. This side here is hyperbolic sine of the angle,
15:41
and this side here is hyperbolic cosine. So everything is exactly the same except for a sine change, and that sine change manifests itself in a sine change in the relation between cosines and sines.
16:01
OK, now we have x equals rho cosh omega, t equals rho sinh omega, almost the same as the equation for x and y up there. Let's guess what the metric looks like. Here it is, ds squared, which is just, where is it?
16:25
Did I write it? Here it is. ds squared is minus dt squared plus dx squared is, is easy to work out, is equal to rho squared d omega squared
16:42
with a minus sign and with a plus sign d rho squared. Let's just compare it. Go up to the metric over here. ds squared is rho squared plus rho squared d rho squared plus rho squared d theta squared. d rho squared, same thing, and instead of rho squared d theta
17:02
squared, we have a minus sign, rho squared d omega squared. Omega is like theta. But there is a big difference between omega and theta. Theta is a periodic variable. It goes around on a circle. It goes from zero to two pi, and it never gets bigger than
17:21
two pi, and then of course it jumps back to zero again. Omega goes from minus infinity way down here, goes up to here, becomes zero at this point, and then goes up to plus infinity. So hyperbolic angles, unlike ordinary angles, go from minus infinity to infinity.
17:43
And of course you can see that. What happens, what happens when omega gets very large? Let's think about for a moment, what happens when omega gets very, very large? What happens to hyperbolic cosine of omega? It's very easy. Just look at this picture. It gets big.
18:02
In fact, it gets big exponentially fast with omega. So this gets very big. What about cinch? Cinch is another way of talking about hyperbolic sine. What happens to it? Does it get big or small? No, it's big. Yeah, how do I know? If this gets very big, and the difference between cosh and cinch is only a tiny number one,
18:28
then cinch better get as big as cosh, apart from the difference one. So if cosh is 100 billion, then cinch also has to be 100 billion minus a little tiny bit. So when omega gets large, cosh and cinch become very close to being equal to each other.
18:47
All right, that means when you go way out at large hyperbolic angle, you're way out here, way up on this light cone here, where t and x are almost equal to each other.
19:02
Way out here on this hyperboloid. So way out on this hyperboloid is where omega gets large. Omega goes to infinity way out on the hyperbola, and way down here it goes to minus infinity. We're going to see that omega is a kind of time variable. If it wasn't clear already, we're going to see that omega is a kind of time variable that keeps time,
19:25
but it keeps time in this angular kind of way. Move along these hyperboloids, or hyperbolas, omega is a kind of time, rho is a kind of distance, and that's the metric.
19:43
The s squared is minus rho squared, the omega squared plus, we haven't worked this out, but it follows in a very similar manner to the way that it did up there. Okay, so here's something. Let's keep this in mind. I want you to remember this formula. In fact, I'm not even going to trust you to remember it. We're going to leave it on the blackboard.
20:02
And we're going to come back to the Schwarzschild metric. Let's come back to the Schwarzschild metric. Oh, let me just say it now. This metric with this peculiar rho dependence here, what space is it? What space time is it?
20:20
It's just flat Minkowski space with nothing going on special anywheres on it. It's just we've chosen to write it in polar coordinates where the coordinate system has something funny going on at the origin, but there's nothing funny physically going on there. It's just a coordinate gimmick.
20:43
Okay, what we want to show now is that this metric close to the horizon is basically like this. It takes a couple of steps. I wish I could do this all in one quick step, but it takes a couple of steps. But they're not hard.
21:01
They're just on the edge of what I like doing on the blackboard. Okay, let's rewrite the metric once. Let's multiply, let's write this as r minus 2mg over r.
21:21
I've just written 1 as r over r dt squared minus, and this is plus r over r minus 2mg times dr squared.
21:43
This coefficient and this one are inverse to each other. I've done nothing. I've simply rewritten it in a form which a little neater. Okay, now what I'm interested in is the black hole very near the horizon.
22:00
In other words, r very close to 2mg. What we want to do is we want to zoom in on r equals 2mg, and sort of expand out and see what's going on there by making some approximations,
22:24
but approximations which are highly accurate right at r equals 2mg. And the basic approximation we're going to make is in the denominator here, we're not going to be moving much. Here's, let's suppose, here is r equals 2, this is r, r axis.
22:42
Here is 2mg. We're going to only be moving a little bit just to explore the very, very neighborhood of this point. So we're going to make little excursions away from this point, but r is not going to change much.
23:01
So I'm going to replace r by 2mg. Should I do the same over here? Now that's not a good idea, because notice what happens when you move r a little bit near here. This thing can change sign, so we better keep it the way it is. Better not fool around with it.
23:21
The 2mg, the denominator here, that doesn't do anything very interesting as you move r a little bit. If 2mg is a kilometer, and you move a centimeter, 2mg only changes a tiny bit, a kilometer plus or minus a centimeter. But r minus 2mg, it flips sign.
23:44
That's dangerous, so let's keep this just the way it is. All right, likewise here, I've just turned this thing over and put it here. So that's the first step, to make a little bit of an approximation. That's the first step.
24:00
Haven't done much. Now, the next step is to look at this over here and say, look, this is just the rho squared over here. I'm going to do a trick. I'm going to change variables. I'm going to change coordinates from r to a new coordinate, which I'll call rho, okay?
24:23
In such a way that this whole thing here, and we'll see how to do this in a moment, in such a way that this whole thing here is just called the rho squared. Now, how can I do that? How can I say that r minus 2mg, whatever it is here, oh, sorry, this should be 2mg here, right?
24:45
2mg, 2mg in the numerator. Just how can I do that? Well, I haven't told you what rho is yet. So until I tell you what rho is, there's no contradiction. There just may be that rho is related to r in such a way that this makes sense.
25:02
So let's see if we can figure out what the relationship between rho and r is. That's the first step. We're just dealing with this term here. Well, it says that the rho squared must be the same, which is another word for equals, equals 2mg over r minus 2mg, the r squared.
25:26
Let me take the square root of both sides. Square root, square root, square root. This is an equation that determines rho in terms of r. How do I find rho in terms of r from an equation like this?
25:43
You integrate both sides of the equation, all right? So what do you get on the left-hand side if you integrate the left-hand side? Rho. What do we get on the right-hand side?
26:02
That's a little harder, huh? Not too much harder. Let's take out the fact that 2mg. And now this becomes dr over square root of r minus 2mg. This piece is not, that's just a constant.
26:23
Anybody smart enough to do that integral? It takes me about 10 minutes. I'm going to wait for somebody else to do it. 2 times r square root of r minus 2mg. 2? I think I heard you say it, right? Times the square root of r minus 2mg.
26:43
This integral over here is 2 times r square root of r minus 2mg. That's the integral of this. So let's write the correct equation. Rho is equal to square root of 2mg times another factor of 2.
27:08
That makes this 8 times square root of r minus 2mg.
27:20
Okay, if you don't believe it, differentiate rho with respect to r and check. This is the same equation as saying that the rho by dr is equal to 2mg over r minus 2mg. Here's the solution. Differentiate with respect to r and reproduce that.
27:41
That's an exercise for you. All right, so now we've found what rho has to be in order that we can take this whole mess here and just write drho squared. Isn't that nice? We now don't have to worry about any factors multiplying this, but it may be getting a little complicated because
28:01
we're going to want to rewrite everything in terms of rho. We're changing variables. And here r appears nice and simply, right? Well, it's not so bad. Look at this. R minus 2mg times something is equal to rho squared.
28:23
Square this equation. Let's square it. Let's see, maybe I should write it over. Yeah. Rho squared is equal to 8mg times r minus 2mg.
28:45
Holy smoke, I now know what r minus 2mg is. It's just rho squared over 8mg. That wasn't so bad. And now we can write the metric r minus 2mg.
29:01
Let's even go a little further. Let's divide by another 2mg here because that's what we're going to need. And we're going to get 16m squared g squared. 16m squared g squared. As I said, a little bit tedious, but not too bad. Okay, so let's see what we have here. We have rho minus rho squared dt squared divided by 16m
29:29
squared g squared plus this. Well, take a look at this. First look at this.
29:40
And then look at this. They're not the same. But they're fairly close. You have a d rho squared. And you have a rho squared times a dt squared. But you'll have this nuisance 16m squared g squared downstairs, right?
30:01
That doesn't appear over here. Well, it's really easy to get rid of it. How do you do it? We've changed the r variable to rho. Now we're going to change the t variable to omega.
30:23
I don't think we need what's in the bottom here. We've done all of that. Now just say that t over 4mg, call that omega.
30:41
dt squared over 16m squared g squared is d omega squared. It's just a change of variables by rescaling the time by this factor 4mg. And what do we get then? We get that this is equal to minus rho squared d omega
31:00
squared plus d rho squared. In other words, we get exactly flat space in hyperbolic polar coordinates. Now, it wasn't exact because we did make a slight
31:21
approximation. But it's an approximation which was essentially exact at r equals 2mg. We replaced an r by a 2mg here and an r by a 2mg here. But that's okay because we're only moving r by a tiny amount away from 2mg.
31:42
So what have we seen? We've seen that to a high approximation, the region near the horizon, near r equals 2mg, has a metric. Where is it? Well, it has a metric which is just the metric of flat space.
32:01
This is good enough to tell us that there's nothing fancy going on at the horizon, nothing traumatic, no forces become infinite. Nothing traumatic happens to somebody who finds himself there. It's just a change of coordinates.
32:21
And there's a c squared. And c squared is all over the place. Can I ask a question about the rho and d rho? If you put the equation back on the board, you expect that for small d rho's when you integrate, rho won't have
32:44
radical changes. But if r is slightly larger than the Schwarzschild radius, it's positive. If it's equal, rho is 0. And if it's negative, rho is i, is an imaginary number.
33:04
It's like an odd integral. Does that have any effect? Well, we've just reproduced what's here. Same thing, the same. No. Yeah.
33:21
Okay. Let's remind ourselves what these coordinates look like. They're drawn over here, but let me redraw them. They're simply Minkowski space, a light cone, an origin, a hyperbolic angle that measures time, so to speak,
33:44
along there. But really, when we put it all together, it's just flat space and some funny coordinates. What's this point right over here in terms of r?
34:04
That point is, of course, rho equals 0. That point is, of course, rho equals 0. But what does rho equal 0? Rho equals 0 is r equals 2mg. So you can imagine somebody coming in from far away, is far
34:24
away from the black hole, moving in, moving in, moving in, moving in, and getting to that point. And that point is the horizon of the black hole. That's where the horizon is, right at that point.
34:41
Now we can start to understand a little better. It's a little weird. We start with a black hole, which looks like a, well, what does it look like? Far from the black hole, it looks just like flat space. Close to the horizon, it also looks like flat space, but
35:03
in a very, very different way. The time up there becomes the hyperbolic angle times 4mg. So it's a rather odd transformation. But the main point here is that nothing special is going on at this horizon over here. OK, any questions about this?
35:24
OK, let's talk about where the horizon is now. Somebody who is outside the horizon is at an r which is larger than 2mg. What does that mean in terms of rho?
35:40
It means at a positive value of rho squared. It means out here. Remember, the hyperbolas are lines of constant rho. Here's rho equals one value, here's rho equals another
36:01
value, here's a bigger value of rho, and so forth. Outside the black hole is to the right of this pair of lines. That's outside the black hole. If you're outside the black hole, let's suppose over here, and you want to send a message which will continue to
36:25
move outward, you can send the message out, and that message will eventually pass any one of these hyperbolas. That message, a light ray.
36:41
You send the light ray out, and that light ray will cross any one of these hyperbolas. So if we imagine a person stationed at each one of these hyperbolas, that means stationed at a value of r. At each value of r, we have another guy. They're standing still, which means they're on a fixed r
37:04
trajectory, which means a fixed rho trajectory. Somebody in here sends a message. Can they get a message to somebody far out? Yes, they can get a message to anybody, no matter how far out, because this message will eventually cross every one
37:20
of these hyperbolas. Is one hyperbola a particular proper time or a particular distance? One hyperbola is a given proper distance from the horizon. Distance from the horizon. Distance. Distance. r. Yeah. Right. Each hyperbola is a given distance from...
37:42
Right. Remember, there's a coordinate transformation between r and rho. Where is it? We had it written down somewhere. Here it is. Let's get it back again.
38:02
Oh, I lost something, but it's got a square root and then another square root of 8 mg. Yeah. Okay. As long as r is bigger than 2 mg, the square root is
38:23
positive. If the square root is positive, rho squared is positive. And for each value of rho, there's one value of r. And for each value of r, you can define one value of rho. Let's not worry about negative values for the square root. Each value of rho comes with a value of r.
38:41
So, a given value of r is on one of these hyperboloids. On one of these rho lines. Okay. So, that's the facts for outside the black hole. If somebody is stationed outside the black hole, they can send a message to anybody else, no matter how far out they
39:02
are. No obstruction if you're outside the horizon to communicating with anybody else.
39:21
I have a question. Yeah. I don't understand where the third on the top left panel. I don't understand where the third equation came from the second. I'll now quote you back. That was not a question. So, now you left me.
39:40
Stay here. Stay here. Seems like the third equation doesn't follow from the second. Third equation, one, two, three. Here, from here to here? Yeah, especially the first term. It's got the comment on the right-hand side. r very close to two energies. Oh, okay. If I put r here, then does it follow?
40:05
As I said, this was the approximation that we made, which is an accurate approximation and is more and more accurate as r becomes closer and closer to mg.
40:22
Where are we? Yeah. All right. Now, what about if you're over here? Can you send a message out to here? Remember, light moves on 45 degree trajectories. Light moves on light cones.
40:42
If somebody is over here, can they send a message out to the outside? The outside meaning anywhere is out beyond this light cone? No. They can't. So, the world sort of divides into the place where you can send a message out beyond this light cone or where you can
41:04
send messages out to people at positive values of r and those places where you can't. That's the definition of a horizon. In fact, this whole light-like line is the horizon.
41:24
If you fall through here, let's now imagine two friends, Alice and Bob. All right. Bob insists on staying outside the black hole.
41:41
And he's going to stay at a fixed distance from the black hole. So, there he is. He's going along on his trajectory. And Alice decides to fall past that point. Once Alice passes this point over here, she can no longer
42:03
send a message to Bob. Bob can't see her. If he can't get a light ray from her, he can't see her. He can't get a message from her. Can Alice see Bob? Yeah. No problem. Here's Alice over here. Alice looks back on her light cone and sees Bob.
42:24
So, there's an asymmetry. Alice can see Bob, but Bob can't see Alice. Alice is past the horizon of the black hole. But does anything happen at that point? No. Nothing special at this point.
42:41
This was just good old flat space and nothing special happened. Yeah. That's because the photons are going from Bob toward Alice so that she can see him. Yeah. But if she jumps through the horizon, is in hip deep,
43:01
can she see her own feet? Can she see her own feet? As she passes through. Yeah. Well, if she's outside and her feet pass. Yeah, she can't see her own feet while she's outside. But how long does she stay outside?
43:20
Not very long. As a matter of fact, you can't see your feet right at this instant. You can only see your feet after this little time for light to get to your eye from your feet. By that time, Alice's eyes see the horizon. But there's a continuous view of her feet as she falls
43:41
through. There's like a little blip where. No, no, no. Nothing at all. Nothing at all. She's just, she's doing what she always does. This is not a special place. Now, should her head, after her feet pass through the horizon, should her head decide, oh no, I am not going to
44:03
pass through the horizon, then her head will never see her feet. That's if she is going to move along one of these trajectory and her feet move into here.
44:20
We don't want, this is not a good thing. This is not a good thing. And then her feet, yeah. So, as long as you don't do anything violent, what you see when you fall into the black hole is what you would have seen if you hadn't fallen into a black hole.
44:43
What somebody sees of you, if that somebody is stationary with respect to the R coordinates, that's a different story. So, let's see what a person, stationary, let's put them
45:00
on one of these trajectories. Is there anything unusual about one of these trajectories? They're curved.
45:21
What does it mean that they're curved? That they're accelerated. They're accelerating. Should this be a surprise that to stay out of the black hole, somebody has to accelerate? Are you accelerating now?
45:42
Yep. At least relative, you're accelerating relative to a freely falling frame. Right. Exactly. Right? So, to stay out of the black hole, it's true that Bob has to accelerate. Maybe he has to have a rocket.
46:02
I would, if I didn't have the floor here to support me, I would have to have a jet pack to keep from falling down. Right? Okay. So, nothing terribly unusual. The further away you get, the less intense the acceleration has to be. But it is true that Bob has to accelerate to stay outside.
46:21
He lets go of Alice. He lets go of Alice and Alice falls into the black hole. He's got the jet pack. She lets go of her hand and she falls in, let's say over here. She doesn't accelerate, so she moves on a straight line. So, there is an asymmetry. There's a basic physical asymmetry that Bob has the jet
46:42
pack on. Alice doesn't. And Alice falls behind in the horizon over here. She can continue to see Bob. Nothing special happens. Bob loses track of her. Cannot see her. So, let's see what Bob can see. Here's Bob.
47:01
What Bob can see is light coming to him. Light comes to him along these light cones. So, when he's over here, he sees Alice over here. Alice is waving to him over here. You go a little bit further, he sees Alice over here. Go a little bit further, he sees Alice over here.
47:25
What happens asymptotically as he looks back from his position at increasing omega? His omega is getting larger and larger and larger. And he's looking back. Does he ever see Alice fall through the black hole horizon?
47:42
No. Ultimately, asymptotically, when Bob looks from the arbitrary future, he simply sees Alice getting closer and closer and closer to the horizon, but never crossing it. So, from Bob's perspective, Alice does not cross the
48:01
horizon. She just gets closer and closer and closer to the horizon, sort of getting pancaked onto it, if you like. It's not just her nose that gets flattened to the horizon. The back of her head does, too. Here's her nose. Here's the back of her head, following her in.
48:21
And they both, he looks back and he sees them both extremely close to the horizon. So, he sees asymptotically, they get closer and closer, but never quite falling through that horizon here. What about Alice? Does she say, does she find any obstruction at this point?
48:41
No. No obstruction at all. She just sails through happily. Nothing happens to her. So, there's already, just at this level, not even from black holes, just thinking about accelerated observers who stay outside of this trajectory, outside of this light cone here,
49:01
there's already a bit of a tension. An observer who moves on a trajectory like this, on a hyperbola, has a set of observations that include among them that Alice falls toward the horizon,
49:20
but never sees her go through the horizon. Indeed. Well, then we have to worry about quantum mechanics. Remember, Planck length has H-bar in it. And we cannot answer that question without some quantum mechanics. Now, we've already done a little bit of quantum mechanics
49:41
about black holes. They're very elementary. We derived the entropy and some other things about the black hole by saying we fill it up with photons. If you remember what we did, there was an H in the formula. So, we've already started to think about the quantum mechanics
50:02
of black holes, but from a different perspective. If you look at rho as a function of R, if you take a big R and R gets smaller, it'll go along the X horizon until it hits that crossing point when R equals...
50:21
Let's draw the difference here. Let's draw first R equals, little R. Little R here equals one. Here's R equals one. That could be one kilometer. Everywhere's along there. Now, let's draw... Sorry, this is big R, little R, excuse me. Little R, one kilometer from the horizon.
50:41
So, R is bigger than two mg. Two mg plus a kilometer. Two mg plus a kilometer. Here's two mg plus half a kilometer. Here's two mg plus a quarter of a kilometer. Here's two mg plus an eighth of a kilometer.
51:02
Right on here. Oh, then it's in here somewhere, but we don't want it to go there yet. For the moment, we don't want to go there. So, rho, the integral, it basically takes a right angle at that crossing point. Yeah, that's right. Does that mean it's not,
51:21
you can't take its derivative at its right angles? Well, something singular happens at that point, but it's not, the main thing is it's not a physical phenomenon that takes place. It's a change of coordinates. It's a funny glitch in the coordinates. And we'll come to it. We'll talk about what's behind here and how you think about it in coordinates, but for the time being,
51:41
we have everything we want just by drawing these pictures. Is omega a proper time in the accelerated reference? Almost. Almost. What is a proper time is rho times omega. Okay, remember the metric for,
52:01
where is it? The metric for omega is rho squared d omega squared, right? So, that means rho, along one of these trajectories, fixed rho, for fixed rho, rho times d omega is proper time along there. So, it's almost, it's proper time,
52:22
but with the rho scaled out. It's the same thing as talking about circles. Is angular proper distance along a circle? Well, not quite. It's proper distance, except you have to multiply by the radius to make the proper distance.
52:40
Okay. All right. That's the properties of a black hole horizon. And the bottom line is that the properties of a black hole horizon are boring. Well, they're boring to somebody who falls through, but they're kind of interesting to somebody who stays outside.
53:00
Somebody who stays outside sees their friend do some weird things, sees them slow down, sees them get flattened against this horizon. And so, there's a kind of tension there, but so far no paradox, just a tension in the description of things. You know, at this point you could ask, well, does Alice really fall through,
53:22
or doesn't she really fall through? Well, what's the answer? The answer is it's a question of point of view. I mean, from Bob's point of view, no, she doesn't. From Alice's point of view, she doesn't. I have a suggestion. Yeah. The answer is that Bob can only see finitely far into Alice's future.
53:40
Why is that? Oh, finitely far into Alice's future. That's correct. That's what Alice would say. Bob says, I see her forever and ever. She just gets slower and slower and duller and duller, and then she just does it. No, but I mean, we make this unconscious adjustment to see our feet as in being a common part of their eyes,
54:00
so we have the same unconscious, you know, kind of... Forget unconscious. The question of, it's physics, it's a question of measurement and observation. Right. So Bob's future doesn't have sight lines into Alice's entire future from her point of view. Bob will simply say, what I see is Alice approaching
54:25
the horizon asymptotically and never passing it. That's what he'll say. You can tell him a story, well, Alice really fell through, blah, blah, blah. He will still... He won't see her age. He won't see her age.
54:41
She... No, she just... Everything slows down. Everything slows down, and she... But it's not a profitable question to ask, does this or that really happen? It is a profitable question to ask what observations and what measurements and correlation of data
55:02
Bob will do as a physicist or as whatever he is, and we could ask what kind of things Alice detects and observes. And that's all we can do. Alice sees Bob like zipping away. And accelerating... She's seen through.
55:21
Yeah, that's right. Alice sees Bob's way. Could you say the part again about that this isn't specific to black holes? Is this any horizon? I think you said something that this isn't specific to black holes. This phenomenon is general. Right. This particular phenomenon would be just for a uniformly
55:43
accelerated observer. If we had an observer riding on a rocket that had enough fuel to allow it to accelerate indefinitely, moving away, then that observer on that rocket would
56:01
say for all practical purposes there was an end to the world, could not see beyond it, and could see nothing fall through it. So he would say there is a horizon. It's called an acceleration horizon. But it's not physically different than the black hole horizon.
56:20
Remember what Einstein taught us. He said if you want to understand gravity, first understand acceleration. If you want to understand how phenomena happen in a gravitational field, first understand how they happen in a uniformly accelerated frame of reference. Well, what we've learned is that in a uniformly
56:40
accelerated frame of reference, and that's what this is, the sequence of lines here, there is an acceleration horizon. So Einstein could have invented the idea of a horizon before there was any notion of a black hole. And then if he was smart enough, if he was as smart as Einstein, he would have said, well,
57:03
this probably means that there's some gravitational context in which these horizons show up. Einstein incidentally did not believe in black holes. He thought they violated something.
57:21
Okay. Now I want to turn back. Before we do, we'll take a break. But I want to turn back to the discussion that we started last time about temperature, entropy a little bit,
57:41
and the thermodynamic properties of black holes that they have because they do have a temperature. So we'll do that after a seven-minute break. Let's see. Maybe I'll get away with one blackboard.
58:01
We'll see. Let me put on my thermodynamic hat now. Heat. What we did last time is we imagined building up the black hole by dropping little drops in, the same way we would drop drops into the bathtub
58:21
and fill it up. And try to see how many drops it takes to fill it to a certain level. In the black hole case, it was how many bits or how many photons each carrying one bit of information do we have to put in to create a black hole of a given size. And what we discovered is that each bit that we drop
58:44
in increases the area of the horizon by one Planck length. When I say one, take one with a grain of salt. We didn't do the real calculation, which gives a factor of a quarter. Why a factor of a quarter?
59:01
Well, what it probably means, what it does mean is that whoever invented the idea of the Planck length, and it was Planck, simply got it wrong by a factor of four. He should have defined it to be four times as big or one-fourth, I forget. And then we wouldn't have this nuisance four hanging around.
59:23
After all, it was purely a historical time and Planck knew nothing whatever about black holes, horizons, information, black holes and so forth. So we just did some dimensional analysis and said, here's a length. If he were to define it as four times or two times different, then the four wouldn't have been there.
59:42
Okay. One of the things we did was to calculate the change in energy if we drop in one bit. Again, remember what a bit means? A bit means a photon whose wavelength is comparable to the
01:00:00
This whole size of a black hole. How much does the energy change? Now, I also told you last time that the definition of temperature, let's just go back a step. Temperature seems like a naive, not a naive meaning to say an easily understood concept.
01:00:22
You feel it with your fingers on the table here. You measure it with thermometers. We all know what it means, but in fact, it's an extremely subtle and somewhat derived concept. Entropy is an extremely confusing thing.
01:00:40
When people go to learn thermodynamics for the first time, entropy is thrown at them. They say, what the hell is that? In fact, entropy is very, very basic and in some sense, simple. Energy and entropy are really the basis for thermodynamics or for statistical mechanics. Temperature is a derived concept.
01:01:01
What is entropy? Well, whatever it is, it comes in bits. It has to do with information. It has to do with yes, no questions. Energy, let's suppose we all knew what energy was, which I suspect most of us have some idea. So, we have entropy and energy.
01:01:21
The definition of temperature is simply the energy that's liberated, the energy change in the system when you add one bit of information to it, the minimum change of energy when you add one bit. I say minimum.
01:01:41
The nice analog is the computer. You want to erase a bit of information from your computer. That bit of information has to go someplace. It can't disappear. It goes out into the atmosphere and it necessarily adds a certain amount of energy to the atmosphere. Of course, if you are sloppy in erasing the bit, you know,
01:02:03
you do it with a sledgehammer, you can add a lot more energy to the atmosphere than that. But if you're very careful, of course, you'll also be adding more bits of information. But if you're very careful to just erase that bit in a minimal way, then the minimum energy that you have to add
01:02:22
to the atmosphere in that case is called the temperature, the energy necessary to change the entropy by one unit. Okay, so you keep that definition in the back of your head of what temperature is. Energy needed to change or energy liberated when you add one
01:02:42
bit to a system. Okay, let's go back now and say here we have our black hole. We're going to throw in a photon whose wavelength is comparable to the size of the black hole and let's calculate the change in energy.
01:03:01
All we have to know is how much energy one photon of a wavelength, here's the wavelength, the wavelength is comparable to the Schwarzschild radius, let's call it r, it's equal to twice the mass times g, that's also the wavelength of the photon, looks to me like my wavelength
01:03:20
is a little bit long, let's shorten it up a little bit, something like that. So what's the energy of a photon of wavelength lambda? Lambda is the usual definition of the wavelength, usual symbol for it. What's the energy of it?
01:03:41
The energy of it is Planck's constant divided by lambda and now there's a speed of light and the speed of light is in the numerator. For the moment I'm going to keep the constants. Later on today, tonight we're going to throw away the constants or not throw them away but we'll work in units in which they're equal to one. I'll just, I get bored carrying them around but let's
01:04:05
keep them for the time being. Alright, h times c divided by lambda, let's just check if it makes sense. The energy of a photon when Planck's constant is, goes to zero, gets smaller and smaller, that makes sense.
01:04:22
A single photon has a tiny bit of energy, why does it have a tiny bit of energy? Because Planck's constant is so small. It's also true that as the wavelength gets long, the energy of a photon goes down. Long wavelength photons have low energy, short wavelength photons have high energy and the speed of light, you can
01:04:41
just check that it's really got to be there. Okay, that's the energy that you add when you add one photon which carries one bit. Well, the energy to add one bit, that's the temperature. So this must be the temperature of the black hole.
01:05:00
And it is. When I say it's the temperature, I mean within a factor of maybe this twos and pis because we're not keeping track of detail, the numerical constants. I'll tell you the numerical constants later if you want. The temperature is hc over lambda.
01:05:23
Oh, since I'm keeping track, for the moment since I'm keeping track of constants, I've left something out of here. C squared. Speed of light squared, yeah. Alright, so that's the temperature but remember that lambda is supposed to be chosen equal to the size of the black
01:05:41
hole. So let's put that in. Either R or hc over twice mg and another c squared in the numerator. So Planck's constant times c cubed and we can't trust this
01:06:03
factor of two. The factor of two is not trustworthy. I'll tell you what the right answer is in a minute. Hc cubed divided by the mass times g. Notice one curious thing that as the mass gets bigger and bigger, oh, several things.
01:06:21
First thing, h appears there. That means that the temperature of the black hole is a quantum effect. It would be zero if h bar of Planck's constant was zero. So it's a thing which only happens because Planck's
01:06:41
constant is not zero. It's another way of saying it's there because of quantum mechanics. C cubed, well, it's there for dimensional reasons. The mass is in the denominator. That's interesting. The bigger the mass of a black hole, the colder it is.
01:07:02
Now, that's a little bit odd for the following reason. Mass is energy, e equals mc squared. So energy and mass are the same thing. You're used to the idea when you add energy to something, it gets hotter. A black hole gets cooler when you add energy to it.
01:07:21
The bigger its mass, the cooler it is. That's got an odd consequence. We'll come to that odd consequence in a moment, but let's just keep track of it. This is the temperature of the black hole. I'll give you the right formula now. If I write it in terms of h bar, then I know the answer and it's 8 pi.
01:07:41
That's the actual correct answer. H bar times c cubed divided by 8 pi times mass times Newton's constant. Okay, first question. How hot is a real solar mass black hole? So for that we would stick in a mass, what's the mass
01:08:02
of a solar mass, 10 to the 30th kilograms or something. The big number downstairs, g is a small number, c cubed is huge, and h bar is teeny, teeny, teeny. So there's some competition of big numbers and small numbers, but the final answer is that this is a small number.
01:08:24
This is, but not that small. Well, it's small. This is about, for a solar mass black hole, for a black hole of stellar mass, this is about 10 to the minus 8 Kelvin. So it's colder than anything, let's see, is it colder
01:08:40
than anything in the, I think people talk about getting down to nanoKelvins for some reasons or other, so maybe things in laboratories are actually colder than this. But this is damn cold, okay. It's so cold that it's much, much colder than empty
01:09:01
space, even in the remotest regions of the universe where there are far, far from any stars or anything else, we know that the temperature of empty space, to the extent that empty space, that space is empty, ordinary empty space is about 3 degrees Kelvin, that's the
01:09:21
microwave background. And so the black hole is, this black hole is colder than anything in empty space. That means even if you had such a black hole, which there was nothing else around to fall into it, even if you managed to create a black hole in otherwise
01:09:43
completely empty space, the empty space around it would be hotter than it is. What happens when you take a cold object and put it in a hot bath? Heat flows from the warm region to the cold region.
01:10:04
Another way of saying it is the black hole simply swallows up the microwave radiation, it just sucks it in. But you can also just think of it as the flow of heat from warm to cold. Okay, what happens now, what happens if I'm in outer
01:10:22
space and I'm cold, but somebody brings a heater and warms me up, I get warmer, right, if I absorb some heat. Now let's think about what happens to the black hole. The black hole absorbs some heat from its surroundings, what happens to its mass if it absorbs some heat?
01:10:41
It increases, its energy increases, but then what happens to its temperature? It goes down, it gets even colder. It gets even colder. What happens when it gets even colder? It even becomes more efficient at absorbing heat. But it absorbs heat and it gets colder.
01:11:00
Catch-22, the more it absorbs heat, the colder it gets. But let's imagine that we had a black hole, I don't know where it came from, which happens to be a little bit warmer than the surroundings. Which way does the heat go? Heat goes from the black hole into the atmosphere.
01:11:22
The black hole loses mass. What happens when it loses mass? It gets hotter. So it's a runaway situation. If a black hole is cooler than its environment, it will absorb energy and get colder.
01:11:41
If the black hole is warmer than its environment, it will give off energy and get hotter, and it will just run away. As it's running away, of course, and getting hotter, its mass is getting smaller and smaller, sort of an explosive situation. Its mass gets smaller and smaller. It gets hotter and hotter, and eventually gets so hot
01:12:01
that it just explodes, if it runs in that direction. So a black hole in thermal equilibrium with its environment is unstable. Even if it had exactly the same temperature as its environment, a fluctuation could happen.
01:12:22
By accident, just by random fluctuation, it might absorb a little bit of extra energy. If it did so, it would get colder. If it got colder, it would be more of an absorber, and it would run away. Or it might accidentally give off a little more extra energy, in which case it would get hotter, and then
01:12:42
give off yet more. So in that sense, black holes are unstable. You might think that means they're unphysical. There are other systems in nature which have this property. In fact, any system that's held together by gravity, a star, stars, what happens to a star when it gives off
01:13:03
energy? It shrinks. That's right. It collapses. When it gives off energy, it thinks it ought to cool. I'm going to cool. Because it thinks it's going to cool, it thinks it has less pressure. The stars don't think, but I need a, you know.
01:13:24
All right, so less pressure to hold the stuff out. Because it has less pressure, stuff starts to fall in. What happens when it falls in? Roughly speaking, you can just imagine that the gas and stuff in that star is getting squeezed. What happens when you squeeze it? It gets hotter. Yeah.
01:13:41
So a star is an example of a system held together by gravity, which when it gives off energy, gets hotter. A star in thermal equilibrium with its environment. Now, thermal equilibrium with its environment would mean that the environment would have to have the temperature of the star. So we're not talking about the real universe, but if we
01:14:02
had a star and we tried to put it in the thermal equilibrium with its environment, it would do the same thing. It would also run away. So you say that was negative specific heat? Negative specific heat. Right. Exactly. Negative specific heat is unstable. But, I mean, it doesn't mean black holes don't exist.
01:14:21
It means, in particular, in the real world, there'll simply be absorbers, which will slowly absorb radiation. Eventually, the universe will expand and cool, and the temperature will go down below the temperature of the black hole, in which case the black hole will start to give off energy and it will evaporate.
01:14:43
Okay. So let's talk about evaporation. Well, yeah. They have temperature, and because they have temperature, they're black bodies. Black bodies not in the sense that they are completely dark, but in the sense that they give off thermal radiation.
01:15:00
I'm going to go through the, well, just a very simple derivation of the rate at which black holes evaporate, the rate at which they lose their energy, but I'm going to do it in units. Since I really don't want to try to remember all the places where C, H bar, G come into this, we'll work
01:15:22
in units where everything is one. And then we'll convert back to sensible units. Now, like I always like to say, if I were on a desert island and I didn't have a physics book and I didn't have a computer or anything else, what units would I work in? They would be Planck units because those are the only
01:15:42
units that I really remember things in. It's helpful to have some rough idea of how big things are in Planck units. And I'm going to go through the derivation of the luminosity of a black hole in Planck units.
01:16:00
As I said, the only reason is everything is one, so you never have to worry where the constants go. Let's remember what Planck units are. They are units in which the speed of light, Planck's constant, and Newton's constant are all equal to one.
01:16:21
C equals H bar equals G equals one. In those units, the unit of length, the Planck length, is about 10 to the minus 35 meters. The Planck time is about 10 to the minus 42, 43
01:16:44
seconds, and the Planck mass, length, time, and mass, that's a complete set of units, and the Planck mass is about 10 to the minus 8 kilograms.
01:17:02
The Planck mass is not a particularly unusual mass. 10 to the minus 8 kilograms is about the mass of a dust mote. You can see it with your naked eye. So it's a rather ordinary mass. The Planck time is incredibly short, and the Planck length is incredibly small. In fact, the Planck time is nothing but the time that
01:17:22
it would take for a light ray to cross the Planck distance. So they're not really independent. If you take the Planck distance and ask how long it would take light to go across it, it's the Planck time. All right, let's work in those units. Good.
01:17:40
Now, a black hole has temperature. Because it has temperature, it radiates. Anything that has temperature, in particular, if it's put into a vacuum, will radiate black body radiation. Black body radiation is simply thermal radiation. Electromagnetic radiation, and incidentally, it will also radiate gravitational radiation, but the theories are
01:18:02
absolutely parallel. All right, let's talk about the formula for the luminosity of a warm body. The luminosity is the rate at which it loses energy. So we can write it as minus only because the object is
01:18:21
losing energy. We can write it as minus the rate of change of the energy of the object. Some object, the edt is the rate of change of its energy. It's losing energy, so de by dt is negative, and minus de by dt is positive.
01:18:45
What is that? First of all, it's proportional to the area of the object. When an object radiates, it radiates from the surface. The bigger the area, the faster it will radiate. So the area of the object appears here, area, and this
01:19:04
will become the area of the horizon of the black hole. Black holes radiate from the horizon. And then, it depends on the temperature. Obviously, the hotter the temperature, the faster it radiates. Anybody know what formula to put here for the temperature?
01:19:22
T to the fourth, famous formula. Stefan-Boltzmann formula. Temperature to the fourth power. You can derive that from dimensional considerations. It's just a dimensional formula. And what about the coefficient which appears here? Order of magnitude.
01:19:40
It's one because all of the parameters are one. So what else could it be? Right. So in Planck units, in Planck units, this is the rate at which the black hole or anything else would lose energy. Now, energy is mass. C equals mc squared, but c is one. So, e is m.
01:20:03
E is m, and we can write this as the rate of change of the loss of mass. The rate of change of the mass of the... Loss of mass per unit time is the area times the temperature to the fourth. What is the area? Well, the area is proportional to the
01:20:22
Schwarzschild radius squared. Let's put that in there. There's some four pi, but by now we don't care about those things. Four pi times r Schwarzschild squared. And what about the temperature? We seem to have lost the formula for the temperature.
01:20:41
Go back in your notes and see if you can find the temperature. Well, h bar is one. C is one. But what's mg? But what's mg? R. Right. Let me just remind you.
01:21:02
The temperature we found was h over lambda times c. This was the energy added when we added one bit. H and c are one. And so the temperature is just one over lambda, but lambda was supposed to be the radius of the black hole.
01:21:22
So the temperature is just one over the radius, the Schwarzschild radius. All right. So temperature to the fourth, that's one over radius to the fourth. And this whole thing just becomes over radius to the fourth. Or the rate of change of mass with time is equal to
01:21:42
one over r squared. Oh, but r is m, isn't it? Yeah. Because r is two mg and g is one. And two is one. Right. So this is also just one over m squared.
01:22:03
Just remember, wherever you see r, you can put m because they're proportional to each other. M is energy. This is the time rate of change of energy. And area one over r to the fourth is temperature to the fourth. So that's all we did.
01:22:21
That's what this formula says. Okay. Let's multiply it by m squared minus m squared dm. What is m squared dm?
01:22:42
Or dm squared dm by dt. M squared dm by dt. Yeah. Apart from a factor of three, m squared times dm by dt is just equal. Oh, what was on the right hand side of this?
01:23:01
There's an equation. What happened to the right hand? It's one, right? One. All right. So another way to write this apart from a factor of three is the time rate of change of the mass cubed of the black hole is equal to one. How long does it take for the black hole to lose all of
01:23:23
its mass? It has a uniform rate at which it loses mass cubed. So its mass cubed is diminishing one unit of it per
01:23:41
unit of time. How long does it take for all that mass that you start with to disappear? Right. M cubed. M cubed. Another way you could write this is the change in mass cubed divided by the change in time is one and therefore the total time that it takes to lose all of the
01:24:00
mass is just m cubed. So m cubed, that's how long it takes. That's the evaporation time. How long is it?
01:24:20
All right. Let's plug in some numbers. Solar mass black hole. What's the solar mass? Ten to the thirtieth kilograms?
01:24:42
So we have to work in Planck units. We did this in Planck units. Let's do it in Planck units. So the solar mass is ten to the thirtieth kilograms. How many Planck masses is that? Ten to the thirtieth. Ten to the thirtieth because the Planck mass is ten to the minus eighth kilograms. So ten to the thirtieth Planck masses is the mass of the sun.
01:25:08
Mass of the sun. The symbol stands for sun. Okay. All right. What's the evaporation time? Well, it's m cubed. So it's ten to, how much, what's thirty-eight times three? A hundred and fourteen?
01:25:22
Yeah. So m cubed is ten to the hundred and fourteen. But ten to the hundred and fourteen what? Planck times. Planck times times t Planck. A Planck time is ten to the minus forty-three seconds. So how many seconds does this correspond to?
01:25:42
Ten to the seventy-one. Ten to the seventy-one seconds. How many years is that? One year is three times ten to the seventh seconds. So divide by ten to the seventh. Ten to the sixty-four years.
01:26:06
What's the present age of the universe? Ten to the tenth years. Right. So this is ten to the fifty-fourth universe ages. Black holes evaporate very, very slowly.
01:26:21
The reason, of course, is they're very, very cold. And you get t to the fourth in them. So even after the universe cools to the point where the black hole will evaporate, it's still evaporating very slowly. All right. Let's take a smaller black hole. How about a black hole of mountain mass?
01:26:44
Anybody have an estimate for a mountain? Ten by ten times ten times one. One hundred cubic kilometers.
01:27:00
One hundred cubic kilometers. So how many kilograms is that? Well, I see ten times ten times one of that stuff.
01:27:20
One hundred cubic square cubic kilometers. So cubic kilometers is ten to the, what, six cubic meters times one hundred ten to the seventh. Ten to the seventh what? Cubic meters. Times kilograms. Let's say the density is three. So four. So it would be four times that. Ten to eight. You're saying a cubic?
01:27:42
The density is three times that of water. Four times that of water. Yeah. Okay. How many cubic meters? Ten to the seventh. Ten to the seventh. But you have to multiply by ten to the three because the cubic meter is about a thousand kilograms.
01:28:00
Yeah. Yeah. So ten to the tenth. Yeah. So ten to the tenth. Let's say ten to the tenth kilograms. Ten to the tenth kilograms is how many planks? Ten to the eighteenth. Ten to the eighteenth. Now three times eighteen.
01:28:22
Ten to the fifty-four. That's ten to the fifty-four Planck times. And now we take out forty-three. Was it ten to the eleventh seconds? Ten to the eleventh seconds. A lot shorter.
01:28:41
What's ten to the eleventh seconds? Ten to the four years. Not terribly long. That's still a pretty big black hole. A black hole of a few kilograms.
01:29:01
Well, a few kilograms, the lifetime is very short. It just goes really fast. Bang. And you can work that out. I don't want to work it out now. Okay. That's the luminosity of a black hole. Wouldn't the size of the kilograms or the black hole be really small?
01:29:20
Yeah. Good. The radius is proportional to the mass, right? Okay. The mass of the sun is ten to the thirtieth. And it corresponds to one or two kilometers.
01:29:40
So that means one kilogram would be ten to the minus thirtieth kilometers. That's pretty small. Right. It's not as small as the Planck length, but it's pretty small. Yeah. It's getting there. Right. But all of these assume an ambient temperature of zero. Yeah. Well, once, of course, not all of these.
01:30:02
Some of these would be. No, I think, let's see. Yeah, I think so. Yeah. That's right. Only if the ambient temperature is cooler than they are. Right. Right.
01:30:20
So that's an interesting question. What mass would the black hole have a temperature which is three degrees of background radiation? I'll leave that for you to figure out. Work that out. What's the size of a black hole? Size, meaning radius, and mass of a black hole which is three degrees.
01:30:46
Okay. I think we've done a bit today. Yeah. Now, what's the mechanism for these photons coming out of it? Well, before we can answer that, we have to answer what's the mechanism for the entropy.
01:31:06
In a certain sense, we're not so much guessing. Most of this is based on very definite calculation. But it's the kind of theoretical work where you put some principles together and you say,
01:31:20
if you believe all the principles, then this is what has to be. The conclusion of it is that there are some hidden, small, numerous degrees of freedom that are not apparent in the standard theory of black holes or in the standard theory of gravity. The understanding of it now is that the standard theory of gravity is in a certain sense like fluid dynamics.
01:31:48
Fluid dynamics describes a smooth flow of a uniform fluid. It can also describe more turbulent flows and so forth. But it does not take into account the atomic structure of fluids.
01:32:03
There are, of course, circumstances where fluids might, the atomic structure might become important. But ordinary fluid flow, the equations of it are effective equations which don't take into account the microscopic structure. You do put entropy into the flow, but you don't ask where that entropy came from.
01:32:25
But just the fact that there is entropy to a fluid tells you there's a hidden microscopic structure. It's a clue. It's a hint that there's something there besides just the smooth flow. There's something underlying it at smaller distances. There are hidden, numerous degrees of freedom.
01:32:42
So, as things stand today, well, as things stood at some point, I think we understand a little more now. But we could say the theory that we've talked about up till now is a strong hint that there's more to the theory of gravity.
01:33:02
There's a micro structure to it. There's more to the theory of black holes than just Einstein's theory. There's something smaller, more numerous that we're not keeping track of in ordinary general relativity. And in that sense, it's a hint. It's a hint of something yet to come. We know a little more about it by now.
01:33:23
But let's wait. Let's wait. We'll talk about what the hidden structures might be. Is saying that the black hole gives off energy, is that the same as saying the black hole is giving off information? Yeah. And so if the black hole grows, sucking in information, and then it shrinks again, giving out energy.
01:33:44
Just got to give it back. So you give back all the same information that went into the black hole in the first place. But you've got to reconcile that with the figure that we drew here that says things fall through the horizon.
01:34:00
How the devil do they get back out? Yeah, that's what I'm trying to do. Tell me about it when you get it. That conservation of information, is that a real fundamental principle? That is the most fundamental.
01:34:21
It's so fundamental that people tend to forget about it. Without it, all of the rules of physics would make no sense. For example, the rules that say that invariance is of various kinds. Time translation and variance leads to energy conservation. It does, only if you have the rule.
01:34:44
I'll give you an example. I'll give you an example right now. Here's an eraser. I slide the eraser along here. Now let's forget the microscopic heat and everything else. Let's suppose we lived in a world where erasers really came to rest because they satisfied equations of motion that made them come to rest.
01:35:03
Rather than the dirty physics of friction and so forth. The real fundamental laws of motion made erasers come to rest. Well, first thing is, there would not be information conservation.
01:35:21
Why? Because every eraser comes to rest, no matter how it started. So it simply forgets how it started. It comes to rest and so it loses the information about where it started. How about energy conservation? There's no energy to be conserved. It just comes to rest. How about momentum conservation? No momentum conservation.
01:35:47
So what is wrong with laws of physics which just have erasers come to rest? They violate some of the basic principles of classical mechanics and quantum mechanics.
01:36:04
But in the end, what they're violating is information conservation. So ideas of energy conservation, momentum conservation, all are conservation. Oh, let me say one more thing. Yeah, come back. This eraser comes to rest, it violates momentum conservation. Does this violate translation invariance, the tabletop?
01:36:24
Imagine the tabletop were uniform and went on forever and ever. And the laws of physics were that erasers come to rest. It doesn't violate translation invariance. So the statement that momentum conservation follows from translation invariance is apparently not true here.
01:36:44
You need something else. The other thing that you need is information conservation. Information conservation plus invariance leads to momentum conservation. So yeah, it's not only fundamental, it's probably the most single fundamental idea in physics.
01:37:02
Is it more fundamental than causality or is it the same? Well, no, it's not the same. Is it more fundamental than causality? No, it's probably at the same level though. It's one of these meta-principles without which you couldn't even begin.
01:37:23
What do you mean by causality exactly? You mean that the... Yeah, yeah. It's a similar kind of principle that... Could be? In your lecture in classical mechanics some years ago, you gave Hamilton's formulation of classical mechanics.
01:37:48
And in that, I recall you saying, I saw it on television, that the state space is, maintains a uniform area. Yeah, that's the principle. Yeah. Okay, in classical mechanics, yes, that's called Liouville's theorem.
01:38:03
That's right. And information conservation in classical mechanics is a statement that the phase space volume doesn't change. That you can't shrink the phase space volume or anything like that. In quantum mechanics, it's called unitarity. And unitarity is a principle that orthogonal states stay orthogonal with time, which means that distinctions are always preserved.
01:38:27
And it's deeply connected with energy conservation. Looking at this from an information theoretic perspective, if you look at the entropy functional, it operates on probability, not microstates.
01:38:50
No, it operates on probabilities for microstates. Yes, but not the microstates themselves. In other words, when the entropy is maximized, wouldn't that indicate the most probable distribution?
01:39:12
That's thermal equilibrium, yeah. For information theory, even if it's not thermal equilibrium, it'll define the PDF.
01:39:25
Okay, that's the definition going back to Shannon's paper in 1948. Again, you're dealing with probabilities of microstates. And the entropy being maximized will indicate the expected value of that, the maximum value of the PDF.
01:39:48
The entropy equals a probability distribution. It defines one. If you look at the functional...
01:40:01
Yes, no, the entropy doesn't define a probability distribution. The probability distribution defines an entropy. Ah, that's interesting because in like information theory, it's flipped. No. Entropy is minus the sum over all configurations of the probability of the i-th configuration times the logarithm of the i-th configuration.
01:40:24
This is one number, and one number can't determine p's. p's determine s. s is given in terms of the p's. But this is taking us far afield. This is taking us off...
01:40:41
Yeah, okay. For more, please visit us at stanford.edu.
Recommendations
Series of 9 media